What a fun summer job! And I totally agree with you, just because you are hanging with the little peeps this summer doesn't mean you have to dress like one! While shorts are always a good choice for summer camp, make sure they provide proper coverage (not too short, not too low) and allow good movement. You can opt for a sporty chic look with a pair of bermudas, a girly knit top and comfy sporty flats.
Another great look would be a pair of trim capris or crops and a feminine blouse in a breathable cool cotton. Keep your footwear simple, comfortable (you will most likely be chasing and being chased by little ones a great deal of the time) and also waterproof!
As far as tote bags go, I love this canvas bag because it has a center divider inside as well as many pockets so it is easy to keep organized. It is also very roomy, and can be tossed into the washing machine as needs be.